Gauss-Bonnet formula, finiteness condition, and characterizations of graphs embedded in surfaces

Let G be an infinite graph embedded in a closed 2-manifold, such that each open face of the embedding is homeomorphic to an open disk and is bounded by finite number of edges. For each vertex x of G, define the combinatorial curvature Phi(G)(x) = 1 - d(x)/2 + Sigma(sigma is an element of F(x)) 1/vertical bar sigma vertical bar as that of [8], where d(x) is the degree of x, F(x) is the multiset of all open faces sigma in the embedding such that the closure (sigma) over bar contains x (the multiplicity of sigma is the number of times that x is visited along partial derivative sigma), and vertical bar sigma vertical bar is the number of sides of edges bounding the face sigma. In this paper, we first show that if the absolute total curvature Sigma(x is an element of V(G)) vertical bar Phi(G)(x)vertical bar is finite, then G has only finite number of vertices of non-vanishing curvature. Next we present a Gauss-Bonnet formula for embedded infinite graphs with finite number of accumulation points. At last, for a finite simple graph G with 3 <= d(G)(x) < infinity and Phi(G) (x) > 0 for every x is an element of V(G), we have (i) if G is embedded in a projective plane and #(V(G)) = n >= 1722, then G is isomorphic to the projective wheel P(n); (ii) if G is embedded in a sphere and #(V(G)) = n >= 3444, then G is isomorphic to the sphere annulus either A(n) or B(n); and (iii) if d(G) (x) = 5 for all x is an element of V(G), then there are only 49 possible embedded plane graphs and 16 possible embedded projective plane graphs.